MathematicalModeling2025 (See html and pdf for more information)

MMC competition

Q1

Baysian+Time series

This means we firstly fix a distribution and then I estimate one parameter using OLS while time-series ARMA for residuals to use more information and then use MCMC to find the other information to estimate variance.

\[\hat{\beta} = \text{argmin} \left( \sum_{i=1}^{n} \left( y_i - \beta_0 - \beta_1 x_1 - \sum_{j=2}^{n} \beta_j x_j - \sum_{j=2}^{n} \beta_j z_j - \sum_{j=2}^{n} \beta_{j+n} (x_j \times z_j) \right)^2 + \lambda \sum_{j=1}^{n} |\beta_j| \right)\] \(x_1\) is Host(0,1)

\(X_j\) is ability(medals/all medals)

\(Z_j\) is chance(participanting program/all programs)

NegativeBinomial inversegamma(not good enough…need further works…)

Q2

Penalized regression model

\[\hat{\beta} = \text{argmin} \left( \sum_{i=1}^{n} \left( y_i - \beta_0 - \beta_1 x_1 - \sum_{j=2}^{n} \beta_j x_j - \sum_{j=2}^{n} \beta_j z_j - \sum_{j=2}^{n} \beta_{j+n} (x_j \times z_j) \right)^2 + \lambda \sum_{j=1}^{n} |\beta_j| \right)\] \(x_1\) is Host(0,1)

\(X_j\) is ability(medals/all medals)

\(Z_j\) is chance(participanting program/all programs)

linear regression for explanation

If we only use each country’s chance(how many this country participating / all programs)*ability(medals this country have in this program / all medals in this program) + Host in each sport program, we have following good-fitting linear relationship, meaning the importance of chance and ability, though it is trivial…:

K-means

Q3